Large cardinals and elementary embeddings of V Master thesis of: Marios Koulakis Supervisor: Athanassios Tzouvaras, Professor à¨å ¢¦¨å£à¤ ¡ §¦¢¦ ©£¦ç ¡ £ ¢∀ ¦ ¡ã - 1997 ©« ¨æ¨ «§«¬® ¡æ ££ To the memory of my parents, Ioannis Koulakis and Anastasia Chrysochoidou Contents 1 Preliminaries 1 1.1 Cumulative hierarchies . 1 1.2 TheMostowskicollapse ....................... 5 1.3 Absoluteness ............................. 6 1.4 Vα and Lα ............................... 12 1.5 Modeltheory ............................. 17 2 Some smaller large cardinals 21 2.1 Cardinalarithmetic.......................... 22 2.2 Inaccessiblecardinals. .. .. .. .. .. .. .. .. .. .. 26 2.3 Mahlocardinals............................ 30 2.4 Weakly compact,Erd˝osand Ramsey cardinals. 33 3 Measurable cardinals and elementary embeddings of V 37 3.1 Aspectsofmeasurability. 37 3.2 Elementary embeddings of V .................... 42 3.3 Normal measures, indescribability . 48 4 0♯ and elementary embeddings of L 53 4.1 Indiscernibles ............................. 53 4.2 0♯ ................................... 54 4.3 The connection of 0♯ with L ..................... 60 4.4 Elementary embeddings of L .................... 62 5 Stronger embeddings of V 69 5.1 Strong,Woodinandsuperstrongcardinals . 69 5.2 Stronglycompactcardinals . 72 5.3 Supercompactcardinals . 73 5.4 Extendible cardinals and Vop˘enka’s principle . 75 5.5 HugecardinalsandI0-I3. .. .. .. .. .. .. .. .. .. 76 5.6 Kunen’stheorem ........................... 78 5.7 Thewholenessaxiom......................... 79 i ii CONTENTS Introduction This thesis is a short survey of the theory of large cardinals. The notion of elementary embeddings is crucial in this study, since it is used to provide a quite general way of defining large cardinals. The survey concludes with a reference to Kunen’s theorem, which sets some limitations to the existence of specific elementary embeddings, and an attempt of P. Corazza to surmount those limitations. Most of the material used through the survey is defined in chapter 1, thus the text is accessible to all readers having knowledge of basic set theory and logic. Chapter 1 serves as an extensive introduction to some notions needed for the rest of the text. Mostowski’s collapsing theorem provides us with a way of transforming a lot of models of set theory, into standard transitive ones. Some results concerning L´evi’s hierarchy illuminate the notion of absoluteness, i.e. the case in which a formula or a term behaves the same way in V as in a standard transitive model. The important cumulative hierarchies Vα and Lα are introduced, along with some of their basic properties. Finally, there is a quick reference to some model theoretical constructions we will use later on. On chapter 2, the first large cardinals appear, though they are not strong enough to yield a connection with elementary embeddings. The first section of this chapter is somehow a continuation of the first chapter, as it provides some fundamental knowledge on cardinal arithmetic. Afterwards, we define in- accessible cardinals, then go on to Mahlo cardinals and finally introduce weakly compact and Erd˝os cardinals. There is an additional reference to Ramsey car- dinals, which have a combinatorial nature similar to that of Erd˝os cardinals, even though they are essentially stronger than the other large cardinals we have mentioned here. On chapter 3 things seem to get more interesting, as we introduce measur- able cardinals. They emerge in a very natural way from measure theory and their study moves from analysis to set theory. They are connected with the elementary embeddings of the form j : V ≺ M, and out of this connection we get the result, due to Scott, that if there is a measurable cardinal, then V 6= L. Concluding, we define normal measures, which become invaluable in the study of elementary embeddings. Chapter 4 goes a little bit backwards, since we study a weaker large cardinal hypothesis, the existence of 0♯. Some effort is needed in order to define 0♯ but it pays back, as 0♯ is very close to the critical question of whether L is a good iii iv CONTENTS approach of V . Silver’s theorem and Jensen’s covering theorem give a more than satisfactory answer to this question which had been roughly approached by measuble and Ramsey cardinals. We finally see, through a theorem of Kunen, that 0♯ is connected with the elementary embeddings of the form j : L ≺ L. Chapter 5 goes on from the point where chapter 3 ended, as it presents the use of elementary embeddings of the form j : V ≺ M, in order to obtain stronger large cardinals. This way, strong, Woodin, strongly compact, supercompact and extendible cardinals appear. We come up with even stronger hypotheses, such as Vop˘enka’s principle, the existence of huge cardinals and I0-I3. The latter, seem to lie near the existence of an embedding j : V ≺ V , which cannot exist due to Kunen’s result, thus there is a probability that they could be disproved from ZF C. Finally, we examine the case of an elementary embedding j : V ≺ V . Kunen’s theorem states that there is not such an embedding. In the process, we examine the consequences of the wholeness axiom, proposed by Corazza, which asserts the existence of an embedding j : V ≺ V that is not weakly definable. The wholeness axiom may weaken the hypotheses of Kunen’s result, but it implies the existence of a super-n-huge cardinal, thus it is very near inconsistency and it should be further investigated. Chapter 1 Preliminaries In this chapter, we present some basic notions of set theory and model theory which will be necessary for our study. On the same time, we try to give some motives for the introduction of large cardinals. Our first step is to present the well known axiomatic system ZF C. Most of the axioms are presented as the closure of V under certain operations, in order to provide a more vivid image of the structure of V . Afterwards, the most powerful of those operations, the powerset operation, is used in order to introduce the notion of cumulative hierarchies. Cumulative hierarchies are roughly sequences of levels Uα, which expand, by the addition of more complex sets, as α gets bigger. Two very common hierarchies are then introduced; von Newmann’s hierarchy Vα, and the hierarchy of the constructible sets Lα. Moving towards a model theoretical study of set theory, Mostowski’s collapsing theorem becomes very useful, as it gives us the opportunity to work, in almost all cases studied here, with standard transitive models. Since standard models share the same relation with V , we are interested in the cases where some formulas have the same truth value in a stander transitive model as in V . The notion that expresses those cases is absoluteness, and an easy way to study it is the introduction of L´evi’s hierarchy of formulas. Next, we meet again the hierarchies Vα, Lα and this time prove some of their basic properties. Finally, we state some model theoretic concepts, such as Skolem functions, ultraproducts and reflection principles. For a more extensive treatment of the material covered in this chapter, the reader is referred to [7], [11] and [21]. For an elementary introduction to set theory, he is referred to [15] and [12]. A classical book on model theory is [2] and much more information on ultraproducts can be found in [1]. 1.1 Cumulative hierarchies The axioms of ZFC The first two axioms describe the nature of sets: 1 2 CHAPTER 1. PRELIMINARIES • A1. Extensionality axiom ∀z (z ∈ x ↔ z ∈ y) → x = y • A2. Foundation axiom ∃y y ∈ x → ∃y ∈ x ∀z ∈ xz∈ / y The next 7 axioms express that the universe V is closed under certain oper- ations, and are written in the form x ∈ V → Q(x) ∈ V where Q denotes an operation. These are the following: • A3φ. Separation axiom x ∈ V →{y ∈ x : φ(y)} ∈ V • A4. Empty set axiom → ∅ ∈ V • A5. Pairing axiom x, y ∈ V →{x, y} ∈ V • A6. Power set axiom x ∈ V → P (x) ∈ V • A7. Union axiom x ∈ V → ∪x ∈ V • A8. Infinity axiom (inf) ∅ ∈ V → ω ∈ V • A9φ. Replacement axiom (F) ′′ x ∈ V → Fφ (x) ∈ V ′′ Where Fφ (x)= y ↔ ∀z (z ∈ y ↔ ∃w ∈ x φ(w,z)) for φ which define functions, i.e. ∀x ∃!y φ(x, y). Finally, the tenth axiom, the axiom of choice, demands the existence of a well-ordering for every set. • A10. Choice axiom (AC) ∀x ∃y y wellorders x 1.1. CUMULATIVE HIERARCHIES 3 An immediate consequence of the axiom of foundation is the fact that there can not exist infinite ∋ sequences. If we combine this with the axiom of ex- tensionality, which states that every set is defined exactly by its elements, we can imagine a construction of V , beginning with the empty set and recursively moving to new sets using axioms A4-A10. We could also add new sets which do not appear in this construction, but may be contained in sets we have already been constructed. We should note here that by ZF C − Ai we mean the axiom system ZF C without the axiom Ai, where 1 ≤ i ≤ 10. We also denote, ZF = ZF C − AC and Z = ZF − F . Cumulative hierarchies The above approach is quite vivid and reveals the structure that V should have, in order to satisfy ZFC. Unfortunately, it is quite complicated, especially when dealing with sets of infinite cardinality. What could be done to simplify it, is to begin with the empty set and produce an ⊂ sequence hUα : α ∈ Oni of sets with increasing complexity, relative to ∈. Using this method, we approach V by constructing bigger and bigger sets of its elements instead of adding them one by one. This idea is captured by the notion of cumulative hierarchies. Definition 1.1. A transfinite sequence hUα : α ∈ Oni is called a cumulative hierarchy if the following are true: (i) U0 = ∅; (ii) Uα ⊂ Uα+1 ⊂ P (Uα); (iii) Uα = β<α Uβ, limit(α).